advertisement

4/23/2011 What is a Hypothesis? Hypothesis Testing Presented by: Mahendra Adhi N A hypothesis is a claim (assumption) about a population parameter: population mean Example: The mean monthly cell phone bill of this city is μ = $42 population proportion Example: The proportion of adults in this city with cell phones is p = .68 Sources: Anderson, Sweeney,wiliams, Statistics for Business and Economics , 6e Pearson Education, 2007 http://business.clayton.edu/arjomand/ Chap 10-2 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. The Null Hypothesis, H0 The Null Hypothesis, H0 (continued) States the assumption (numerical) to be tested Example: The average number of TV sets in U.S. Homes is equal to three ( H0 : μ = 3 ) Is always about a population parameter, not about a sample statistic H0 : μ = 3 H0 : X = 3 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-3 The Alternative Hypothesis, H1 Is the opposite of the null hypothesis Begin with the assumption that the null hypothesis is true Similar to the notion of innocent until proven guilty ilt Refers to the status quo Always contains “=” , “≤” or “≥” sign May or may not be rejected Hypothesis Testing Process Claim: the e.g., The average number of TV sets in U.S. homes is not equal to 3 ( H1: μ ≠ 3 ) population mean age is 50. Challenges g the status q quo (Null Hypothesis: Population H0: μ = 50 ) Never contains the “=” , “≤” or “≥” sign May or may not be supported Is generally the hypothesis that the researcher is trying to support Is X= 20 likely if μ = 50? If not likely, REJECT Null Hypothesis Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-4 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-5 Now select a random sample Suppose the sample mean age is 20: X = 20 Sample Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. © Mahendra Adhi Nugroho, M.Sc, Accounting Program Study of Yogyakarta State University For internal use only! 1 4/23/2011 Level of Significance, α Reason for Rejecting H0 Sampling Distribution of X Defines the unlikely values of the sample statistic if the null hypothesis is true 20 If it is unlikely that we would get a sample mean of this value ... X μ = 50 If H0 is true ... if in fact this were the population mean… ... then we reject the null hypothesis that μ = 50. Chap 10-7 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Is designated by α , (level of significance) H0: μ = 3 H1: μ ≠ 3 α α/2 Two-tail test α/2 Is selected by the researcher at the beginning Provides the critical value(s) of the test H0: μ ≤ 3 H1: μ > 3 Rejection region is shaded h d d α Type I Error Reject a true null hypothesis Considered a serious type of error The probability of Type I Error is α 0 Upper-tail test H0: μ ≥ 3 H1: μ < 3 α Lower-tail test Chap 10-8 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Errors in Making Decisions Represents critical value 0 Typical values are .01, .05, or .10 Level of Significance and the Rejection Region Level of significance = Defines rejection region of the sampling distribution Called level of significance of the test Set by researcher in advance 0 Chap 10-9 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Errors in Making Decisions Chap 10-10 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Outcomes and Probabilities (continued) Possible Hypothesis Test Outcomes Type II Error Fail to reject a false null hypothesis Actual Situation Decision The probability of Type II Error is β Do Not Key: Outcome (Probability) Reject H0 Reject H0 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-11 H0 True H0 False No error (1 - α ) Type II Error (β) Type I Error ( α) Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. © Mahendra Adhi Nugroho, M.Sc, Accounting Program Study of Yogyakarta State University For internal use only! No Error (1-β) Chap 10-12 2 4/23/2011 Factors Affecting Type II Error Type I & II Error Relationship Type I and Type II errors can not happen at the same time Type I error can only occur if H0 is true Type II error can only occur if H0 is false If Type I error probability ( α ) , then Type II error probability ( β ) Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-13 β when α β when σ β when n Chap 10-14 Two Basic Approaches to Hypothesis Testing The power of a test is the probability of rejecting a null hypothesis that is false i.e., β when the difference between hypothesized parameter and its true value Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Power of the Test All else equal, ¾ There are two basic approaches to conducting a hypothesis test: ¾ 1- p-Value Approach, and ¾2- Critical Value Approach Power = P(Reject H0 | H1 is true) Power of the test increases as the sample size increases Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Slide 16 Chap 10-15 1- p-Value Approach to One--Tailed Hypothesis Testing One 2- Critical Value Approach One--Tailed Hypothesis Testing One Use the Z table to find the critical Z value, value, and Use the equation to find the actual ZZ--Z statistics. In order to accept or reject the null hypothesis the p-value is computed using the test statistic ---Actual Actual Z value. Reject H0 if the h p-value l <α The rejection rule is: L Lower tail: il Reject R j H0 if Actual A l z < Critical C i i l -z α • Upper tail: Reject H0 if Actual z > Critical zα • Do not reject (accept) H0 if the p-value > α In other words, if the actual Z (Z statistics) is in the rejection region, then reject the null hypothesis. Equation for finding the actual Z value: z= x −μ σ/ n Slide 17 © Mahendra Adhi Nugroho, M.Sc, Accounting Program Study of Yogyakarta State University For internal use only! Slide 18 3 4/23/2011 Test of Hypothesis for the Mean (σ Known) Hypothesis Tests for the Mean Convert sample result ( x ) to a z value Hypothesis Tests for μ σ Known Hypothesis Tests for μ σ Unknown σ Known σ Unknown Consider the test The decision rule is: H0 : μ = μ0 H1 : μ > μ0 x − μ0 > zα σ n Reject H 0 if z = (Assume the population is normal) Chap 10-19 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Decision Rule Reject H0 if z = p-Value Approach to Testing H0: μ = μ0 x − μ0 > zα σ n H1: μ > μ0 Alternate rule: α Reject H0 if X > μ0 + Z ασ/ n p-value: Probability of obtaining a test statistic more extreme ( ≤ or ≥ ) than the p value g given H0 is true observed sample Do not reject H0 Z 0 x μ0 Chap 10-20 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. zα Reject H0 μ0 + z α Also called observed level of significance Smallest value of α for which H0 can be rejected σ n Critical value Chap 10-21 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-22 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Example: Find Rejection Region p-Value Approach to Testing (continued) Convert sample result (e.g., x ) to test statistic (e.g., z statistic ) Obtain the p-value x - μ0 p - value = P(Z > , given that H0 is true) For an upper σ/ n t il ttest: tail t = P(Z > Suppose that α = .10 is chosen for this test Find the rejection region: Reject H0 α = .10 x - μ0 | μ = μ0 ) σ/ n Do not reject H0 Decision rule: compare the p-value to α If p-value < α , reject H0 If p-value ≥ α , do not reject H0 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. (continued) 0 1.28 Reject H 0 if z = Chap 10-23 Reject H0 x − μ0 > 1.28 σ/ n Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. © Mahendra Adhi Nugroho, M.Sc, Accounting Program Study of Yogyakarta State University For internal use only! Chap 10-24 4 4/23/2011 Example: Upper-Tail Z Test for Mean (σ Known) Example: Sample Results (continued) A phone industry manager thinks that customer monthly cell phone bill have increased, and now average over $52 per month. The company wishes to test this claim. (Assume σ = 10 is known) Obtain sample and compute the test statistic Suppose a sample is taken with the following results: n = 64, 64 x = 53.1 53 1 (σ=10 was assumed known) Form hypothesis test: H0: μ ≤ 52 the average is not over $52 per month H1: μ > 52 the average is greater than $52 per month x − μ0 53.1 − 52 = = 0.88 σ 10 n 64 z= (i.e., sufficient evidence exists to support the manager’s claim) Chap 10-25 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Using the sample results, Example: Decision Example: p-Value Solution (continued) Calculate the p-value and compare to α Reach a decision and interpret the result: p-value = .1894 α = .10 0 1.28 z = 0.88 Reject j H0 α = .10 Do not reject H0 Do not reject H0 since z = 0.88 < 1.28 i.e.: there is not sufficient evidence that the mean bill is over $52 Chap 10-27 H0: μ ≤ 3 = .1894 Chap 10-28 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Upper-Tail Tests In many cases, the alternative hypothesis focuses on one particular direction H1: μ > 3 = P(z ≥ 0.88) = 1− .8106 Reject H0 Do not reject H0 since p-value = .1894 > α = .10 One-Tail Tests 1.28 Z = .88 P(x ≥ 53.1| μ = 52.0) 53.1− 52.0 ⎞ ⎛ = P⎜ z ≥ ⎟ 10/ 64 ⎠ ⎝ 0 Reject H0 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. There is only one critical value, since the rejection area is in only one tail H0: μ ≤ 3 H1: μ > 3 α This is an upper-tail upper tail test since the alternative hypothesis is focused on the upper tail above the mean of 3 Do not reject H0 H0: μ ≥ 3 H1: μ < 3 (continued) (assuming that μ = 52.0) Reject H0 Do not reject H0 Chap 10-26 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. This is a lower-tail test since the alternative hypothesis is focused on the lower tail below the mean of 3 Z x zα 0 Reject H0 μ Critical value Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-29 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. © Mahendra Adhi Nugroho, M.Sc, Accounting Program Study of Yogyakarta State University For internal use only! Chap 10-30 5 4/23/2011 Lower-Tail Tests Two-Tail Tests H0: μ ≥ 3 In some settings, the alternative hypothesis does not specify a unique direction H1: μ < 3 There is only one critical value, since the rejection area is in only one tail α Reject H0 Do not reject H0 -zα Z 0 x μ Critical value H0: μ = 3 H1: μ ≠ 3 α/2 There are two critical values, defining the two regions of rejection α/2 x 3 Reject H0 Do not reject H0 -zα/2 Lower critical value Chap 10-31 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Reject H0 z +zα/2 0 Upper critical value Chap 10-32 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Hypothesis Testing Example Hypothesis Testing Example (continued) Test the claim that the true mean # of TV sets in US homes is equal to 3. (Assume σ = 0.8) State the appropriate null and alternative hypotheses H0: μ = 3 , H1: μ ≠ 3 (This is a two tailed test) Specify the desired level of significance Suppose that α = .05 is chosen for this test Choose a sample size Suppose a sample of size n = 100 is selected Determine the appropriate technique σ is known so this is a z test Set up the critical values For α = .05 the critical z values are ±1.96 Collect the data and compute the test statistic Suppose the sample results are n = 100, x = 2.84 (σ = 0.8 is assumed known) So the test statistic is: z = Chap 10-33 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. X − μ0 σ n = 2.84 − 3 − .16 = = − 2.0 0.8 .08 100 Chap 10-34 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Hypothesis Testing Example Hypothesis Testing Example (continued) Is the test statistic in the rejection region? Reject H0 if z < -1.96 or z > 1.96; otherwise do not reject H0 α = .05/2 Reject H0 -z = -1.96 α = .05/2 Do not reject H0 0 (continued) Reject H0 Reach a decision and interpret the result α = .05/2 Reject H0 +z = +1.96 -z = -1.96 α = .05/2 Do not reject H0 0 Reject H0 +z = +1.96 -2.0 Since z = -2.0 < -1.96, we reject the null hypothesis and conclude that there is sufficient evidence that the mean number of TVs in US homes is not equal to 3 Here, z = -2.0 < -1.96, so the test statistic is in the rejection region Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-35 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. © Mahendra Adhi Nugroho, M.Sc, Accounting Program Study of Yogyakarta State University For internal use only! Chap 10-36 6 4/23/2011 t Test of Hypothesis for the Mean (σ Unknown) Example: p-Value Example: How likely is it to see a sample mean of 2.84 (or something further from the mean, in either direction) if the true mean is μ = 3.0? Convert sample result ( x ) to a t test statistic Hypothesis Tests for μ x = 2.84 is translated to a z score off z = -2.0 20 σ Known α/2 = .025 P(z < −2.0) = .0228 .0228 P(z > 2.0) = .0228 Consider the test .0228 H0 : μ = μ0 p-value H1 : μ > μ0 = .0228 + .0228 = .0456 -1.96 -2.0 σ Unknown α/2 = .025 0 1.96 2.0 Z Chap 10-37 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. The decision rule is: Reject H 0 if t = (Assume the population is normal) x − μ0 > t n -1, α s n Chap 10-38 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Example: Two-Tail Test (σ Unknown) t Test of Hypothesis for the Mean (σ Unknown) (continued) For a two-tailed test: Consider the test H0 : μ = μ0 (Assume the population is normal, and the population variance is unknown) k ) H1 : μ ≠ μ0 The decision rule is: x − μ0 x − μ0 Reject H 0 if t = < − t n -1, α/2 or if t = > t n -1, α/2 s s n n Chap 10-39 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. The average cost of a hotel room in New York is said to be $168 per night. i ht A random d sample l of 25 hotels resulted in x = $172.50 and s = $15.40. Test at the α = 0.05 level. (Assume the population distribution is normal) Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Example Solution: Two-Tail Test H0: μ = 168 H1: μ ≠ 168 α = 0.05 0 05 Reject H0 -t n-1,α/2 -2.0639 σ is unknown, so use a t statistic Critical Value: t24 , .025 = ± 2.0639 α/2=.025 Do not reject H0 Involves categorical variables Two possible outcomes Reject H0 0 1.46 x−μ 172.50 − 168 t n −1 = = s 15.40 n 25 t n-1,α/2 2.0639 = 1.46 Do not reject H0: not sufficient evidence that true mean cost is different than $168 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-40 Tests of the Population Proportion α/2=.025 n = 25 H0: μ = 168 H1: μ ≠ 168 Chap 10-41 “Success” Success (a certain characteristic is present) “Failure” (the characteristic is not present) Fraction or proportion of the population in the “success” category is denoted by P Assume sample size is large Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. © Mahendra Adhi Nugroho, M.Sc, Accounting Program Study of Yogyakarta State University For internal use only! Chap 10-42 7 4/23/2011 Proportions Example: p-Value (continued) Sample proportion in the success category is denoted by p̂ x number of successes in sample pˆ = = n sample size Here: When nP(1 – P) > 9, p̂ can be approximated by a normal distribution with mean and standard deviation P(1− P) μp̂ = P σ p̂ = n Chap 10-43 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. If p-value < α , reject H0 If p-value ≥ α , do not reject H0 p-value = .0456 α = .05 α/2 = .025 Since .0456 < .05, we reject the null hypothesis H0 False 1.96 2.0 Z Chap 10-44 Assume the population is normal and the population variance is known. Consider the test Do Not Reject H0 No error (1 - α ) Type II Error (β) Reject H0 Type I Error ( α) No Error (1-β) H0 : μ = μ0 H1 : μ > μ0 Th decision The d i i rule l iis: Reject H 0 if z = β denotes the probability of Type II Error 1 – β is defined as the power of the test x − μ0 > z α or Reject H0 if x = x c > μ0 + Z ασ/ n σ/ n If the null hypothesis is false and the true mean is μ*, then the probability of type II error is ⎛ x −μ* ⎞ ⎟ β = P(x < x c | μ = μ*) = P⎜⎜ z < c σ / n ⎟⎠ ⎝ Power = 1 – β = the probability that a false null hypothesis is rejected Chap 10-45 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-46 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Type II Error Example 0 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Actual Situation H0 True Decision .0228 Type II Error Recall the possible hypothesis test outcomes: Key: Outcome (Probability) α/2 = .025 .0228 -1.96 -2.0 Power of the Test (continued) Compare the p-value with α Type II Error Example Type II error is the probability of failing to reject a false H0 Suppose we fail to reject H0: μ ≥ 52 when in fact the true&x&c mean is μ μ* = 50 (continued) Suppose we do not reject H0: μ ≥ 52 when in fact the true mean is μ* = 50 This is the range of x where H0 is not rejected Thi iis th This the ttrue distribution of x if μ = 50 α 50 52 Reject H0: μ ≥ 52 Do not reject H0 : μ ≥ 52 xc Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. 50 Reject H0: μ ≥ 52 Chap 10-47 52 xc Do not reject H0 : μ ≥ 52 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. © Mahendra Adhi Nugroho, M.Sc, Accounting Program Study of Yogyakarta State University For internal use only! Chap 10-48 8 4/23/2011 Type II Error Example Calculating β (continued) Suppose we do not reject H0: μ ≥ 52 when in fact the true mean is μ* = 50 Here β = P( x ≥ Here, Suppose n = 64 , σ = 6 , and α = .05 ) if μ* x = 50 (f H0 : μ ≥ 52) (for c So β = P( x ≥ 50.766 ) if μ* = 50 β α 50 Reject H0: μ ≥ 52 α 52 xc σ 6 = 52 − 1.645 = 50.766 n 64 x c = μ0 − z α 50 Do not reject H0 : μ ≥ 52 50.766 Reject H0: μ ≥ 52 Chap 10-49 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. 52 Do not reject H0 : μ ≥ 52 xc Chap 10-50 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Calculating β Power of the Test Example (continued) If the true mean is μ* = 50, Suppose n = 64 , σ = 6 , and α = .05 ⎛ ⎞ ⎜ 50.766 − 50 ⎟ P( x ≥ 50.766 | μ* = 50) = P⎜ z ≥ ⎟ = P(z ≥ 1.02) = .5 − .3461 = .1539 6 ⎜ ⎟ 64 ⎠ ⎝ The probability of Type II Error = β = 0.1539 The power of the test = 1 – β = 1 – 0.1539 = 0.8461 Actual Situation Probability of type II error: α β = .1539 50 Reject H0: μ ≥ 52 Do not reject H0 : μ ≥ 52 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. H0 True Do Not Reject H0 No error 1 - α = 0.95 Reject H0 52 xc Key: Outcome (Probability) Decision Type I Error α = 0.05 H0 False Type II Error β = 0.1539 No Error 1 - β = 0.8461 (The value of β and the power will be different for each μ*) Chap 10-51 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-52 Chapter Summary Addressed hypothesis testing methodology Performed Z Test for the mean (σ known) Discussed critical value and p-value approaches to h hypothesis th i ttesting ti Performed one-tail and two-tail tests Performed t test for the mean (σ unknown) Performed Z test for the proportion Discussed type II error and power of the test Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 10-53 © Mahendra Adhi Nugroho, M.Sc, Accounting Program Study of Yogyakarta State University For internal use only! 9